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Journal of the Australian Mathematical Society

Expectation values of operators in the quasi chemical equilibrium theory

John M. Blatt

Published Dec 1, 1960• 18 citations
Publisher

Abstract

Abstract Expectation values of one-particle and two-particle operators are evaluated in the quasi-chemical equilibrium (pair correlation) approximation to statistical mechanics. Certain reductions, corresponding to the “quenching” of interactions by the Pauli exclusion principle, are carried out quite generally. More specific reductions, which lead to immediately useful expressions, are possible on the assumption of extreme Bose-Einstein condensation of the correlated pairs.

DOI
10.1017/s1446788700026264

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Journal of the Australian Mathematical Society•Dec 1, 1962•1 citation

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Progress of Theoretical Physics•Mar 1, 1960•47 citations
Abstract Expectation values of one-particle and two-particle operators are evaluated in the quasi-chemical equilibrium (pair correlation) approximation to statistical mechanics. Certain reductions, corresponding to the “quenching” of interactions by the Pauli exclusion principle, are carried out quite generally. More specific reductions, which lead to immediately useful expressions, are possible on the assumption of extreme Bose-Einstein condensation of the correlated pairs.
The quasi-chemical equilibrium, or pair correlation, approximation to statistical mechanics is written in terms of second-quantization formalism. This involves an entension of the original theory to include an Ansatz for off-diagonal elements of the density matrix. The formal expressions involve “labelling operators” acting in a purely formal Hilbert space. These labelling operators obey Bose commutation rules for correlations between even numbers particles (e.g. pair correlations), Fermi commutation rules for correlations between odd numbers of particles. The labelling operators provide an algebraic way of formulating the restriction to certain types of “graphs”. We also show how to include higher-order correlations into the Ansatz. The main results of the original theory can be derived more rapidly with the new formalism. However, certain correction terms obtained earlier are shown to be in error, necessitating a reinvestigation of the nature of the condensation phenomenon. This re-investigation is carried out in two following papers, and proves to lead to no essential modifications of the previous results. The correlation matrices which enter into the Ansatz need to be related to the Hamiltonian of the system, and to the thermodynamic variables which define its thermodynamic state. A variational formulation is developed for this purpose, but no explicit calculations are carried out in this paper.
In the pair-correlation, or quasi-chemical equilibrium, approximation to statistical mechanics, there appears the possibility of a condensation phenomenon closely analogous to a Bose-Einstein condensation of correlated pairs of particles. However, the expansion used in earlier work to establish this phenomenon fails to yield an adequate approximation at and below the condensation point. In this paper, we give an alternative development, first formally for an arbitrary number of quantum states of the pair, then carry out in detail for the case of only one, or only two, quantum states of the pair. It is shown that, in spite of the Fermi-Dirac statistics of the particles making up the pairs, it is nevertheless possible to accomodate an arbitrarily large number of particles within a single pair state. Furthermore, if two quantum states of the pair are possible, and conditions are such that either the pair state would be occupied by a macroscopic number of particles if it alone were present, then the Pauli exclusion principle works in such a way that the lower one of the two pair states is occupied by a macroscopic number of particles, whereas the occupation number of the upper pair state remains independent of the volume. This result is not restricted to pairs whose mean separation exceeds their internal size, but applies also to pairs which completely overlap each other. The result is therefore of significance in the theory of superconductivity where the electron pairs have an internal size of the order of the Pippard coherence length, which is much larger than the mean separation of their centres of gravity.